cp's OEIS Frontend

A167930 Number of partitions of n in which some but not all parts are equal.

%I A167930 #15 May 30 2024 17:43:21
%S A167930 0,0,0,0,1,3,4,9,13,20,29,43,57,82,110,146,195,258,334,435,558,713,
%T A167930 910,1150,1446,1814,2268,2815,3491,4308,5301,6501,7954,9692,11795,
%U A167930 14295,17301,20876,25148,30200,36218,43322,51741,61650,73354
%N A167930 Number of partitions of n in which some but not all parts are equal.
%C A167930 The parts may not all be equal, and at least one part must occur at least twice. - _N. J. A. Sloane_, May 30 2024
%H A167930 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of the shell model of partitions (2D and 3D view)</a>
%H A167930 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%H A167930 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the shell model of partitions (3D view)</a>
%F A167930 a(n) = A047967(n) - A032741(n).
%F A167930 a(n) = A000041(n) - A000009(n) - A032741(n).
%F A167930 a(0) = 0: For n>0, a(n) = A000041(n) - A000009(n) - A000005(n) + 1.
%e A167930 The partitions of 6 are:
%e A167930 6 ....................... All parts are distinct.
%e A167930 5 + 1 ................... All parts are distinct.
%e A167930 4 + 2 ................... All parts are distinct.
%e A167930 4 + 1 + 1 ............... Only some parts are equal ...... (1).
%e A167930 3 + 3 ................... All parts are equal.
%e A167930 3 + 2 + 1 ............... All parts are distinct.
%e A167930 3 + 1 + 1 + 1 ........... Only some parts are equal ...... (2).
%e A167930 2 + 2 + 2 ............... All parts are equal.
%e A167930 2 + 2 + 1 + 1 ........... Only some parts are equal ...... (3).
%e A167930 2 + 1 + 1 + 1 + 1 ....... Only some parts are equal ...... (4).
%e A167930 1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal.
%e A167930 Then a(6) = 4.
%e A167930 a(7) = 9 from 511  4111  331  322  3211  31111  2221  22111  211111. - _N. J. A. Sloane_, May 30 2024
%t A167930 f[lst_]:=With[{c=Split[lst]},Length[lst]>2&&Max[Length/@c]>1&&Length[c]>1]; Table[Length[ Select[ IntegerPartitions[n],f]],{n,0,50}] (* _Harvey P. Dale_, May 30 2024 *)
%Y A167930 Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167931, A167932, A167933.
%K A167930 nonn
%O A167930 0,6
%A A167930 _Omar E. Pol_, Nov 15 2009
%E A167930 Edited by _Omar E. Pol_, Nov 16 2009
%E A167930 More terms from _Max Alekseyev_, May 02 2011